Hi Russel,
On 15 Feb 2009, at 03:41, russell standish wrote:
>
> On Fri, Feb 13, 2009 at 07:31:29PM +0100, Bruno Marchal wrote:
>>>
>>> I'm a little confused. Did you mean Dp here? Dp = -B-p
>>
>>
>> Fair question, given my sometimes poor random typo!
>>
>
> ...
>> deduce Bp) , well, if you remind the definition of the Kripke
>> semantics, you can see that
>>
>> Bp & Dp
>>
>> is equivalent with
>>
>> Bp & Dt
>>
> ...
>
>> Now if you have in a world, your world if you want, Bp & Dp, you
>> have
>> at least access to a world in which p is true, and thus you have
>> access to a world where t is true, given that t is true in all
>> worlds.
>> So you have Bp & Dt.
>
> Thanks. Alles ist Klar. I think I wasn't taking seriously enough the
> idea of Kripke frames before...
>
> ...
>
>>
>> A good and important exercise is to understand that with the Kripke
>> semantics, ~Dt, that is B~t, that is Bf, that is "I prove 0=1", is
>> automatically true in all cul-de-sac world. It is important because
>> cul-de-sac worlds exists everywhere in the Kripke semantics of the
>> self-reference logic G.
>>
>> If you interpret, if only for the fun, the worlds as state of life,
>> then Bf is really "I am dead".
>>
>> Bruno
>
> Yes, but I have difficulty in _simultaneously_ interpreting logic
> formulae in terms of Kripke frames and B as provability. In the
> former, Bp means in all successor worlds, p is true, whereas in the
> latter it means I can prove that p is true.
>
> How does one reconcile such disparate notions?
By Godel's theorems, Löb's theorems and Solovay theorems.
When B is seen as provability, it really means "I can prove that p is
true" *when* asserted by a self referentially correct universal
machine, believing (or asserting, or proving) the induction axioms,
like Peano Arithmetic, or Zermelo-Fraenkel Set Theory.
In that case the "B" is the corresponding (to PA, ZF, any Lobian
Machine) Gödel provability predicate, and it obeys invariant
mathematical provability laws.
It can be proved that the modal logic G is sound and complete for the
arithmetical provability laws that the machine can prove, and it can
be proved that the modal logic G* is sound and complete for the true
provability laws, with the propositional variable "p" interprete by
closed arithmetical formula. I will (re)come back on this later
probably.
The soundness is a quasi direct consequence of Gödel's and Löb's
incompleteness theorems, together with the fact that the modal rule of
inference preserve the arithmetical modal provability and correctness
(for G and G* respectively).
The completeness is far more complex to prove, and it has been done by
Solovay theorem. (Reference in my Post to Günther, or in my both
thesis where this is explained with various details. The proof of
Solovay is explained in Smorynski book, and in Boolos 1979 and 1993
books in details. It is the base of AUDA, or the "interview of the
Lobian Machine" (see any of my papers or theses).
You get the hypostases by introducing the Theaetical intensional
nuances (Bp & p, Bp & Dp, Bp & Dp & p, etc.). G is really the third
personne self-reference, "Bp & p" gives the first person self-
reference: the real one that the machine cannot even name. Bp & Dp
gives intelligible matter, Bp & Dp & p gives the sensible matter, Etc.
G* knows them equivalent, proving the same arithmetical theorem, but
they obeys veruy different logics due to the fact that the machine
cannot know them equivalent.
And you get the computationalist hypostases by restricting the
interpretation of the propositional variables, p, to the Sigma-1
sentences (which proofs provide the arithmetical Universal Dovetailing).
All this is the substance of AUDA, which leads to the theology
(including the verifiable physics or the logic of the observable) of
the Universal machine.
The corona G* minus G and its intensional variants, give the proper
theological reality of the universal machine. What they can correctly
hope or fear, and correctly bet, without being to prove.
See the second part of the Sane2004 paper for a short but precise
account of AUDA (the first part is UDA(1...8).
See the Plotinus paper for a longer explanation of AUDA, and to see
its use for providing an arithmetical interpretation of both Plotinus
"theory of mind" and "theory of matter", or see the two thesis for
detailed explanation and motivations. Or my old post on "the machine-
itself" and its "Guardian Angel" (if you remind?). Or wait to see if
i will succeed to explain this on the list, or out of the list. I
have already tried with not so much success, but those matter are not
so easy of course. But an understanding of UDA + an understanding of
those mathematical theorems leads "automatically" to AUDA. And to the
discovery that, about machine only, there is already no unifying
complete everything theory. On the contrary you will see that all
universal machine which introspects herself discover or create an
universal Everything *realm* in its own "head" ...
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Sun Feb 15 2009 - 12:41:25 PST